American Mathematical Society

Operator theory, operator algebras, and applications; proceedings.

Papers from a June 2005 symposium are presented here. The symposium was of special interest that year due to its emphasis on applications to subjects outside of mathematics, such as physics, engineering, and theoretical computer science. Most of the papers present original research that has not been published elsewhere. Themes include wavelets and quantum theory, operator theory and harmonic analysis, non-commutative geometry, C* algebras, Von Neumann algebras, frames in Hilbert space, the Kadison-Singer conjecture, extensions and dilations, and multivariable operator theory. Some specific topics include diagonals of self-adjoint operators, trace-class perturbations and functional calculus, and rank preserving maps on CSL algebras. There is no subject index. (Annotation ©2006 Book News Inc. Portland, OR)

Painlevé transcendents; the Riemann-Hilbert approach.

Painlevé's work in second order nonlinear ordinary differential equations included finding that the location of the possible branch points and essential singularities of their solutions does not depend on local conditions. This applies to only six equations, the well-known Painlevé I-VI. However few, these equations are now applied to statistical physics, fluid mechanics, random matrices and orthogonal polynomials, and the solutions of the equations play a significant role in nonlinear mathematical physics. In the 16 chapters here the authors emphasize the connection of the Riemann-Hilbert method the classical monodromy theory of linear equations as well as to modern theories of integrable systems. They cover isomonodromy methods and special functions, including the Backlund transformations, describe a case study for asymptotics of the Painlevé II transcendent, including the canonical six-rays, and examine the asymptotics of the Painlevé III transcendent, including the Sine-Gordon reduction and canonical four-rays. (Annotation ©2006 Book News Inc. Portland, OR)

Problems on mapping class groups and related topics.

The study of mapping class groups, the moduli space of Riemann surfaces, Teuchmuller geometry and related areas has an advantage in its number of experienced scholars and at the same time the increasing number of those new to the field. To get them closer together, this collection of papers focuses on open problems and directions. Topics include cohomological combinatorial and algebraic structure, including finiteness and Torelli spaceshomomorphisms, and probing with arcs, connections with 3-manifolds, symplectic geometry and algebraic geometry, including problems in Heegaard distance and Lefschetz pencils, geometric and dynamical aspects, including problems on billiards and Weil-Petersson perspectives, and braid groups and other related groups, including Iwahori-Heck algebras and free Abelian groups. Papers include exercises and references. (Annotation ©2006 Book News Inc. Portland, OR)

Quantum graphs and their applications; proceedings.

Considered as one-dimensional varieties and equipped with differential rather than difference operators, quantum graphs have applications in a variety of scientific and mathematical fields, including nanotechnology, optics, and number theory. This proceedings volume contains 21 contributions from international researchers discussing the subject from various disciplinary perspectives. A sampling of topics includes form factor expansion for large graphs; small diffusion asymptotics for exit problems on graphs; and the quantum graph as a limit of a network of physical wires. The volume is not indexed. (Annotation ©2006 Book News Inc. Portland, OR)

Recent advances in differential equations and mathematical physics; proceedings.

These proceedings from the March-April 2005 conference include new material and recent surveys on a variety of topics related to mathematical physics. Aimed at postgraduate students but also of interest to many mathematicians and physicists, the contents include surveys articles on Dirichlet-Neumann inverse problems on manifolds, numerical investigations association with Laplacian eigenvalues on planar regions, Snell's law and propagation of singularities in the Wave equation and random operators on tree graphs. Topics also include asymptotic formulas for the ground-state energy of fermionic gas, renormalization ideas in quantum field theory from perturbation so of the free Hamiltonian on the circle, J-selfadjoint Dirac operators, spectral theory of Schrodinger operators, inverse problems, isoperimetric inequalities in quantum mechanics, Hardy inequalities and non-adiabatic transitions. (Annotation ©2006 Book News Inc. Portland, OR)

Recent developments in algebraic topology; proceedings.

This collection of papers is based on talks at a December 2003 conference. Papers illustrate the versatility of algebraic topology interacting with others areas of mathematics and physics. Topics discussed range from classical differential topology and homotopy theory to more recent lines of research such as topological quantum field theory (string theory). Alternative viewpoints on classical problems such as global analysis and dynamical systems are developed. Material reflects the work of Sam Gitler, a pioneer in homotopy theory. There is no subject index. (Annotation ©2006 Book News Inc. Portland, OR)

Representations of algebraic groups, quantum groups and Lie algebras; proceedings.

Papers from a July 2004 conference survey current research on the geometric and combinatorial aspects of algebraic groups, quantum groups, and Lie algebras. New developments in quiver representation are presented in connection with important constructions for quantum groups. Another major theme is the use of methods from algebraic geometry, via derived categories, to study the representation theory of algebraic groups and Lie algebras, including Kac-Moody Lie algebras, modular restricted Lie algebras, finite group schemes, and Lie superalgebras. There is no subject index. (Annotation ©2006 Book News Inc. Portland, OR)

Representation theory, dynamical systems, and asymptotic combinatorics.

Fourteen papers from a June 2004 conference are collected here. Some topics considered are stochastic dynamics related to the Plancherel measure on partitions, skew products and Livsic theory, and transverse properties of dynamical systems. Other subjects examined include the Plancherel measure on shifted Young diagrams, results and problems in enveloping algebras arising from quantum groups, asymptotic behavior in the time synchronization model, stable densities and operators of fractional differentiation, and the distribution of ergodic sums for hyperbolic maps. The audience for the book includes graduate students and researchers in combinatorial and dynamical aspects of group representation theory. There is no subject index. (Annotation ©2006 Book News Inc. Portland, OR)

Steenrod squares in spectral sequences.

Singer develops a general theory with special attention to the change-of-rings spectral sequence for the cohomology of an extension of Hopf algebras and to the Eilenberg-Moore spectral sequence for the cohomology for the sequence for the cohomology of classifying spaces and homotopy orbit spaces. The result is effective in computing the cohomology rings of groups and Hopf algebras, and the Steenrod algebra in particular. He covers conventions, the spectral sequence of a bisimplical coalgebra, bialgebra actions on the cohomology of algebras, extensions of Hopf algebras, Steenrod operations in the change-of-rings spectral sequence, the Eilenberg-Moore spectral sequence, and Steenrod operations in the Eilenberg-Moore spectral sequence. (Annotation ©2006 Book News Inc. Portland, OR)